I have been set to do this exercise, and up until the following questions I was fine:

For reference, the answers are meant to be
6) 2/7*pi, 0.1m, (0.2+0.1sin7t)m
7) 0.018m, 0.359s
8) 7.04m and 9m, 1.99s

For 6, I worked out the tension in the spring (acting on the sphere) to be T=98x (x being the compression), however when substituted into the acceleration = (angular velocity)^2*extension formula, I'm having trouble eliminating x (I end up with g(1-5x = acceleration)).

For 7, I'm not sure what I'm doing wrong. I get the tension in the string to be 49/32 when the extension is 0.05, and then that can be substituted into the acceleration formula, however it ends up with angular velocity = 10.5 which doesn't give the required time period. I have a feeling I should leave the extension as x here, but same as in 6, I have trouble eliminating it at that point.

For 8, I'm really not too sure how I should split the tensions and treat the strings separately - is it just meant to be a ratio of what forces the full string would have, but split? (which doesn't seem to work out).

Any help is appreciated, I feel like I'm making a lot of mistakes with the equations.

(Original post by DFranklin)
One of us is confused: In question 6 there is a spring, not a string. It's under compression, not tension. And you talk about a variable x that I can't see mentioned anywhere in the question.

I would go (sketch only):

Let x be the compression of the spring from it's neutral (unweighted position). We know the spring will exert an upwards force kx upon the ball, for some constant k.

The net force on the ball is 2mg-kx (acting in the same direction as we are measuring compression), and we know this = 0 when x = 0.2. So we can find k.

Note that we can then rewrite 2mg-kx in the form (0.2-x)k.

Finally, observe if we define y = x-0.2 we have and so we have simple harmonic motion. Then you can use standard equations etc.

The string and spring thing I go muddled up with (the equations were correct though) and for x I meant the extension. I got there in the end, but it wasn't easy. Onto 7 and 8 now